The paper presents a compartment model of the dynamics of the incidence of the new coronavirus (COVID-19). An approach to the construction of SIR models is shown. The SIRF model for the COVID-19 epidemic process in Ukraine has been built. Morbidity data provided by the Center of Public Health of the Ministry of Health of Ukraine. The morbidity was analyzed, the assessment of the stay of people in public places before and during quarantine was carried out. Based on the simulation results, the predicted incidence of COVID19 in Ukraine for 10 and 60 days was calculated. The research implementation results enhance the efficiency of management decisions to ensure the biosafety of the population and the development of scientifically based strategies for anti-epidemic and preventive measures. The results obtained in this study expand the possibilities for making correct decisions by administrators who determine strategies in the health care of countries.
its borders and captured most of the countries of the world [
Health systems in most countries were overwhelmed. There were not enough hospital beds, beds in
devices, and other medical supplies [
Patient treatment protocols were often revised and improved, which affected both the patient's
The development of vaccines, which are obtained using different technologies, has become an
additional tool for mitigating the epidemic. Vaccination began in Israel, European countries, the USA,
etc. [
2021 Copyright for this paper by its authors. measures will reduce disease severity and the case fatality rate? The answers to these and other questions have not yet been received.
All forecasting models have different initial assumptions and use historical data differently. Models
based on well-grounded theoretical understanding and available evidence are critical to formulating
viable observational policies, but shifts in distribution can lead to systematic false predictions [
As a matter of fact, COVID-19 has followed specific patterns which are basically related to the
dynamic contagion of the pandemic. When the pandemic occurs, different methods of surrogation
measures are implemented to detect and evaluate these infective diseases [
It is impossible to get answers to all questions, to correctly assess and predict the development of the situation from only observations. Mathematical modeling can help, on the one hand, identify and understand the main driving forces behind the spread of the disease and, on the other hand, evaluate the most effective measures to contain a pandemic for the specific conditions of a particular country, city, area [16], and also enable governments and health systems promptly to provide healthcare institutions with the necessary resources, to develop business tactics, the existence of the population, and to correct preventive and control measures [17].
The research aims to develop a compartment model of the COVID-19 epidemic process using statistical data of COVID-19 in Ukraine.
The SIR (Susceptible, Infected, Recovered) model [18] is the basis for describing the spread of infectious diseases and was proposed in the 1920s by Scottish epidemiologists Anderson Kermack and William McKendrick. According to SIR model, the population is divided into three groups: susceptible ( ), infected ( ) and those who have recovered ( ). Over time, transitions → (infection) and → (recovery or death) are possible. Today there is a whole family of models developed on the basis of SIR models. The SIR-based models are based on the idea that the studied population is divided into compartments (clusters) and assumptions about the nature and rate of transmission from one compartment to another. Diseases that confer immunity have a compartmental structure different from diseases without immunity, and are most often described by ordinary differential equations (which are deterministic), but models with a stochastic (random) structure, which are more realistic, but much more difficult to analyze, can also be used [19].
The three categories are interrelated with one another and with parameters due to the following equations: Rate of change of Susceptible Population is given by:
where β is the pathogen contagion rate and γ is the recovery rate.
The developed model allows us to investigate the dependance of COVID-19 epidemic process dynamics and antiepidemic countermeasures. The purpose of such experiments is to find out which factors influence the dynamics the most.
We have expanded classic SIR model with state “F”. SIR-F parameter estimation is applied to dynamics of COVID-19 epidemic process in time subsets to determine the effects of measures. To investigate infection rate and recovery rate S-R trend analysis has been provided.
Analysis of the incidence of the new coronavirus shows that all infected areas can be divided into two clusters: with a growth factor greater than one, and a growth factor less than one. In the developed model, the growth factor was calculated using the following formula. in which C is the number of confirmed cases.
The rapid spread of the disease, the congestion of hospitals and the lack of information about the new coronavirus infection lead to the death of some patients before the diagnosis of the disease. Therefore, the number of deaths is added to those infected: “S + I → Fatal + I. And SIRF model is described by following equations: (5) in which S is susceptible people, S∗ is confirmed and uncategorized cases, I is infected and categorized cases, R is recovered people, F is dead because of infection, α1 is mortality rate of S∗, α2 is mortality rate of I [1/min], β is contact rate [1/min], γ is recovery rate [1/min], The model can be described by ordinary differential equation (6) (6)
For non-dimensional model we set (S,I,R,F)= N×(x,y,z,w) and (T,α1,α2,β,γ) = (τt,θ,τ−1κ,τ−1ρ,τ−1σ). This model can be described as (7). Reproduction number can be defined as
Now we perform Susceptible-Recovered trend analysis for actual data of COVID-19 incidence in Ukraine. The plot of the S-R trend is shown in figure 2.
We assume that morbidity by novel coronavirus infection in Ukraine have four dates of its dynamics changes.
Now we investigate COVID-19 data in Ukraine (figure 3). Data for further simulation was given by the Center of Public Health of the Ministry of Health of Ukraine.
Let’s apply dates of COVID-19 dynamics changes to our model, and build SIRF model started from the date each stage starts. Figure 4 shows different phases of the COVID-19 epidemic process in Ukraine.
Now we compare developed model results for estimating SIR-F parameters (fig. 5).
A national lockdown was implemented in Ukraine from 13th of March, 2020. A national lockdown affected on gs and c. Let’s assume that people started to meet each other less with the lockdown for 19%. Now we estimate gs before 13th of March (fig. 6).
Simulation shows that the average time people go out without lockdown is 5,8 days. Now we estimate gs after national lockdown (fig. 7). Simulation shows that after lockdown average time susceptible people go out is 0,8 days.
Based on SIR models, we saw that more parameters can be included and we are capable to do more complicated calculations spontaneously by the easiness of implementing and operating on the actual datasets as the input. That means the complexity of SIR models didn’t disturb the easiness of applying it, also it gave a more reliable forecasting output which was exactly what we expected. These models have the ability to forecast the trend not only based on single parameters but including a mix of them and considering the interactions of some and illustrating the result through diagrams and tables. In the last part of implementing this model, we saw that we could have a view of the effect of new medicines on the trend as an external parameter. This shows that SIR models have the capability of adding more parameters as the input and showing their influence on the output which might be practical for further studies. Thus, the results have more credit to rely on for making further investigations and even decisions.
Our research has shown that the SIR model can be used to assess the contribution of various factors to the development of the epidemic. On the one hand, experiments help to understand and analyze the main drivers of the epidemic, for example, the intensity of communication between people, their crowding, etc., which makes it possible to develop and implement measures aimed at eliminating or weakening this driver. On the other hand, experiments with the model make it possible to evaluate the effectiveness of a particular preventive measure aimed at mitigation of the epidemic. For example, what percentage of the population should be vaccinated in order to reduce the reproductive number, which categories of the population should be vaccinated in the first order to reduce the incidence of the population, the effectiveness of masks, the effectiveness of hand rub, etc. The results obtained in this study expand the possibilities for making correct decisions by administrators who determine strategies in the health care of countries.
Based on the predictions we have developed using the approach we have developed, healthcare administrators can timely set the required number of hospital beds, medical personnel, ventilators, and other necessary resources. It should be taken into account that in experiments with the model it is possible to estimate when, for what period of time and in what volume one or another measure should be introduced. For example, how effective will the weekend quarantine, or adaptive quarantine, or complete knockdown be, if it is entered in a specific period of time.
The study was funded by the National Research Foundation of Ukraine in the framework of the research project 2020.02/0404 on the topic “Development of intelligent technologies for assessing the epidemic situation to support decision-making within the population biosafety management” [20].
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